Define a fibonacci series generated by {a,b} such as,

$\displaystyle u_1=a$

$\displaystyle u_2=b$

$\displaystyle u_{n+2}=u_{n+1}+u_{n}$

(The original fibonacci series is generated by {1,1})

($\displaystyle a\not =0$)

Then there are a number of interesting properties:

1)If $\displaystyle K$ is the finite continued fraction for $\displaystyle b/a$

then $\displaystyle \frac{u_{n+1}}{u_n}=[1;1,1...,1,K]$

Where the $\displaystyle 1$ appears $\displaystyle n-2$ times.

2)Thus, from here we have that $\displaystyle \lim_{n\rightarrow \infty}\frac{u_{n+1}}{u_n}=\psi$

(Thus, any fibonacci series generated by any two real numbers converges to the divine proportion).

3)The formula for $\displaystyle u_n$ is given by

$\displaystyle u_n=F(n-2)a+F(n-1)b$

where $\displaystyle F(n)$ is the n-th fibonacci number.

But by Binet's formula we have that,

for the n-th fibonacci number we can find a formula for $\displaystyle u_n$ but it is rather messy and will be omitted. Giving a second method for proving statement 2.